Integer Optimization and Lattices
نویسنده
چکیده
• Lattices. We will see basic properties of lattices, followed by Minkowski’s Theorem which guarantees that any symmetric convex set with volume bigger than 2 must have an non-zero integer point. We will show an application of Minkowski’s theorem to Diophantine approximations. Then we will introduce the powerful concept of Lattice basis reduction which finds an almost orthogonal basis for a lattice in polynomial time. This can be used to obtain (the only known) approximation algorithm for the Shortest vector problem. One application lies in factoring polynomials with rational coefficients. Another surprising application that we will see lies in breaking Knapsack cryptosystems.
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